Which statement is true about the Maclaurin series?

Which statement is true about the Maclaurin series?

• A. Maclaurin series is the Taylor series about a = 0
• B. Maclaurin series is the Fourier series
• C. Maclaurin series is a Legendre series
• D. Maclaurin series uses imaginary numbers only

Answer: (A)

The Maclaurin series is the Taylor series centered at zero. If a function has derivatives at zero, you can write it as a power series around x = 0: f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + … . This shows how the function is approximated by polynomials using information from derivatives at zero. For example, e^x becomes 1 + x + x^2/2! + x^3/3! + …, and sin x begins as x − x^3/3! + x^5/5! . This concept is distinct from Fourier series, which decompose periodic functions into sines and cosines, and from Legendre series, which use Legendre polynomials. It also isn’t limited to imaginary numbers; for real-valued functions the coefficients are real.

The Maclaurin series is the Taylor series centered at zero. If a function has derivatives at zero, you can write it as a power series around x = 0: f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + … . This shows how the function is approximated by polynomials using information from derivatives at zero. For example, e^x becomes 1 + x + x^2/2! + x^3/3! + …, and sin x begins as x − x^3/3! + x^5/5! . This concept is distinct from Fourier series, which decompose periodic functions into sines and cosines, and from Legendre series, which use Legendre polynomials. It also isn’t limited to imaginary numbers; for real-valued functions the coefficients are real.